Fibonacci in Matrix world

Given,

$A=\begin{bmatrix}F_k & F_{k-1}\\0 & 0\end{bmatrix} \cdot \begin{bmatrix}F_k & -F_{k-2}\\-F_{k+1} & F_{k-1}\end{bmatrix}$

$A=\begin{bmatrix}F_k^2-F_{k-1}\cdot F_{k+1} & F_{k-1}^2-F_{k-2}\cdot F_k\\0 & 0 \end{bmatrix}$

Now, we will consider a property of Fibonacci sequence-

$F_n^2-F_{n-1}\cdot F_{n+1}=(-1)^{n+1}$

Therefore, $A=\begin{bmatrix}(-1)^{k+1} & (-1)^{k}\\ 0 & 0 \end{bmatrix}$

Trace of A= $tr(A)=(-1)^{k+1}$

Determinant of A= $|A|=0$

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A property of $2\times 2$ matrix-

$A^2-tr(A)\cdot A+|A|\cdot I=0$

$\implies A^2=(-1)^{k+1}A$

Given, $A^2=A$ , by comparing,

$(-1)^{k+1}=1$

$k+1 \rightarrow even$

$k \rightarrow odd$

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Number of odd natural numbers less than 100 is 50.

But $k \ne 1$ as $F_0$ is not specified.

So, answer is $\boxed{49}$